Subgroups — #16807108

Order 3328.

524 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131210	C₂	1	4	yes	maximal, complemented
2	#13131206	C₂	1	4	yes	maximal, complemented
3	#13131177	C₂	1	2	yes	maximal, complemented
4	C₂×#11718284	C₂	1	2	yes	maximal
5	#13131059	C₂	1	2	yes	maximal, complemented
6	#13123254	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
8	C₂×#88180	C₄	1	1	yes	characteristic
9	C₂²×#6099	C₂²	1	1	yes	characteristic
10	C₂×#88106	C₂²	1	4	yes	—
11	#11718783	C₄	1	1	yes	characteristic
12	#11718783	C₄	1	1	yes	characteristic
13	#11718783	C₂²	1	1	yes	characteristic
14	#11718783	C₂²	1	1	yes	characteristic
15	M₁₆×Dic₁₃	C₄	1	2	yes	—
16	M₁₆×Dic₁₃	C₂²	1	2	yes	—
17	#11718762	C₄	1	2	yes	—
18	#11718762	C₂²	1	2	yes	—
19	#11718284	C₂²	1	4	yes	—
20	#11718284	C₂²	1	4	yes	—
21	#11718262	C₂²	1	8	yes	complemented
22	#11718261	C₂²	1	8	yes	complemented
23	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₃×M₁₆	—	2	4	no	—
25	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₂²⋊Dic₁₃	—	2	1	no	—
30	#88180	—	2	1	no	—
31	#88180	—	2	1	no	—
32	#88180	C₂×C₄	1	2	yes	—
33	#88180	C₂×C₄	1	1	yes	characteristic
34	#88180	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₁₃	—	2	1	no	—
36	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
37	C₂×#6099	—	2	4	no	—
38	C₂×#6099	C₂×C₄	1	1	yes	characteristic
39	C₂×#6099	C₂×C₄	1	1	yes	characteristic
40	C₂×#6099	C₂³	1	1	yes	characteristic
41	C₂×#6099	C₂³	1	1	yes	characteristic
42	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
43	C₂²×#1055	C₂³	1	1	yes	characteristic
44	#88106	—	2	4	no	—
45	#88106	C₂³	1	8	yes	—
46	C₂×#6092	—	2	4	no	—
47	#88055	C₂×C₄	1	2	yes	—
48	#88055	C₂×C₄	1	2	yes	—
49	#88055	C₂×C₄	1	2	yes	—
50	#88055	C₂³	1	2	yes	—
51	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
52	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
53	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
54	C₈×Dic₁₃	C₂³	1	2	yes	—
55	#33427	—	13	1	no	maximal, Sylow
56	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×Dic₁₃	—	2	2	no	—
66	C₁₃×M₁₆	—	2	2	no	—
67	C₁₃×M₁₆	—	2	4	no	—
68	C₁₃×M₁₆	M₁₆	1	4	yes	—
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
72	C₂²⋊Dic₁₃	—	2	2	no	—
73	C₂²⋊Dic₁₃	—	2	1	no	—
74	C₂²⋊Dic₁₃	—	2	1	no	—
75	C₄⋊Dic₁₃	—	2	2	no	—
76	C₄⋊Dic₁₃	—	2	1	no	—
77	C₄⋊Dic₁₃	—	2	1	no	—
78	C₄×Dic₁₃	—	2	2	no	—
79	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
80	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
81	#6099	—	2	2	no	—
82	#6099	—	2	2	no	—
83	#6099	—	2	2	no	—
84	#6099	—	2	2	no	—
85	C₂×#1055	—	2	2	no	—
86	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
87	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
88	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
89	C₂×#1055	#41	1	4	yes	—
90	C₂×#1055	C₂⁴	1	1	yes	characteristic
91	#6092	—	2	8	no	—
92	#2873	—	13	1	no	—
93	#2166	—	13	4	no	—
94	#2165	—	13	4	no	—
95	#2153	—	13	2	no	—
96	C₂×#349	—	13	2	no	—
97	#2110	—	13	2	no	—
98	C₂³×C₁₃	—	2	2	no	—
99	C₂³×C₁₃	—	8	1	no	—
100	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
102	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	#117	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×Dic₁₃	—	2	2	no	—
117	C₂×Dic₁₃	—	2	2	no	—
118	C₂×Dic₁₃	—	8	2	no	—
119	C₈×C₁₃	—	2	2	no	—
120	C₈×C₁₃	C₂×M₁₆	1	4	yes	—
121	#1055	—	2	1	no	—
122	#1055	—	2	1	no	—
123	#1055	#117	1	2	yes	—
124	#1055	C₂×#41	1	2	yes	—
125	C₂²×M₁₆	—	13	1	no	—
126	C₂²×M₁₆	#261	1	1	yes	characteristic
127	C₂×#117	—	13	1	no	—
128	C₂×M₃₂	—	13	4	no	—
129	#405	—	13	1	no	—
130	#405	—	13	1	no	—
131	#405	—	13	1	no	—
132	#405	—	13	1	no	—
133	#405	—	13	2	no	—
134	#405	—	13	2	no	—
135	C₄×M₁₆	—	13	2	no	—
136	C₄×M₁₆	—	13	2	no	—
137	#364	—	13	8	no	—
138	#349	—	13	4	no	—
139	#349	—	13	4	no	—
140	#347	—	13	8	no	—
141	C₂²×C₁₃	—	2	1	no	—
142	C₂²×C₁₃	—	2	1	no	—
143	C₂²×C₁₃	—	4	1	no	—
144	C₂²×C₁₃	—	4	1	no	—
145	C₂²×C₁₃	—	4	1	no	—
146	C₂²×C₁₃	—	8	2	no	—
147	C₂²×C₁₃	#432	1	2	yes	—
148	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	#432	1	2	yes	—
152	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
153	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
154	Dic₁₃	—	4	2	no	—
155	Dic₁₃	—	4	2	no	—
156	C₂³×C₄	C₂×#261	1	1	yes	characteristic
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	13	1	no	—
159	C₂×M₁₆	—	13	1	no	—
160	C₂×M₁₆	—	13	1	no	—
161	C₂×M₁₆	—	13	1	no	—
162	C₂×M₁₆	—	26	4	no	—
163	C₂×M₁₆	C₂×#261	1	2	yes	—
164	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
165	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
166	C₂²×C₈	—	13	1	no	—
167	C₂²×C₈	—	13	1	no	—
168	C₂²×C₈	C₂×#261	1	2	yes	—
169	#117	—	13	2	no	—
170	#117	—	13	1	no	—
171	#117	—	13	1	no	—
172	#117	—	26	1	no	—
173	#117	—	26	1	no	—
174	C₂×C₄⋊C₄	—	26	1	no	—
175	C₂×C₂²⋊C₄	—	26	1	no	—
176	C₂×C₄²	—	13	2	no	—
177	M₃₂	—	13	8	no	—
178	M₃₂	—	26	4	no	—
179	C₂×C₁₆	—	26	4	no	—
180	#97	—	13	2	no	—
181	#97	—	13	2	no	—
182	#97	—	13	2	no	—
183	#97	—	13	2	no	—
184	C₄×C₈	—	13	2	no	—
185	C₄×C₈	—	13	2	no	—
186	C₄×C₈	—	13	2	no	—
187	C₄×C₈	—	13	2	no	—
188	C₂×C₁₃	—	2	1	no	—
189	C₂×C₁₃	—	2	2	no	—
190	C₂×C₁₃	—	4	1	no	—
191	C₂×C₁₃	—	4	1	no	—
192	C₂×C₁₃	C₂×#432	1	1	yes	characteristic
193	C₂⁴	#6118	1	1	yes	characteristic
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	26	2	no	—
197	C₂²×C₄	#6118	1	1	yes	characteristic
198	C₂²×C₄	#6118	1	1	yes	characteristic
199	C₂²×C₄	#6118	1	1	yes	characteristic
200	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
201	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
202	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
203	M₁₆	—	2	2	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	—	26	2	no	—
206	M₁₆	—	26	2	no	—
207	M₁₆	—	26	2	no	—
208	M₁₆	—	26	2	no	—
209	M₁₆	#6122	1	4	yes	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	13	4	no	—
212	C₂×C₈	—	13	1	no	—
213	C₂×C₈	—	13	1	no	—
214	C₂×C₈	—	13	1	no	—
215	C₂×C₈	—	13	1	no	—
216	C₂×C₈	—	26	2	no	—
217	C₂×C₈	C₂²×#261	1	2	yes	—
218	C₂×C₈	C₂²×#261	1	2	yes	—
219	C₄⋊C₄	—	26	2	no	—
220	C₄⋊C₄	—	26	1	no	—
221	C₄⋊C₄	—	26	1	no	—
222	C₂²⋊C₄	—	26	2	no	—
223	C₂²⋊C₄	—	26	1	no	—
224	C₂²⋊C₄	—	26	1	no	—
225	C₄²	—	13	2	no	—
226	C₄²	—	13	2	no	—
227	C₄²	—	26	2	no	—
228	C₁₆	—	26	8	no	—
229	C₁₃	#33427	1	1	yes	characteristic, complemented, Sylow
230	C₂³	—	2	2	no	—
231	C₂³	—	8	1	no	—
232	C₂³	C₂×#6118	1	1	yes	characteristic
233	C₂³	C₂×#6118	1	1	yes	characteristic
234	C₂³	C₂×#6118	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	26	2	no	—
243	C₂×C₄	—	26	2	no	—
244	C₂×C₄	—	104	2	no	—
245	C₂×C₄	C₂×#6118	1	1	yes	characteristic
246	C₂×C₄	C₂×#6118	1	1	yes	characteristic
247	C₂×C₄	C₂×#6118	1	1	yes	characteristic
248	C₂×C₄	#88237	1	2	yes	—
249	C₂×C₄	C₂×#6122	1	2	yes	—
250	C₂×C₄	C₂³×#261	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	13	2	no	—
253	C₈	—	13	2	no	—
254	C₈	—	26	1	no	—
255	C₈	—	26	1	no	—
256	C₈	C₂×#6122	1	4	yes	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	8	2	no	—
263	C₂²	#11719130	1	2	yes	—
264	C₂²	C₂²×#6118	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	52	2	no	—
268	C₄	—	52	2	no	—
269	C₄	#11719130	1	2	yes	—
270	C₄	C₂×#88237	1	1	yes	characteristic
271	C₄	C₂²×#6122	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	2	no	—
274	C₂	—	4	1	no	—
275	C₂	—	4	1	no	—
276	C₂	C₂×#11719130	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#88237
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂²×C₁₃×M₁₆	C₄
socle	C₂×C₁₃	C₂×#432
prime radical	C₂⁴×C₁₃	M₁₆
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂²×M₁₆	#261
hypocenter	C₁₃	#33427
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂²×M₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #13131210	C₂	semidirect
2. #13131206	C₂	semidirect
3. #13131177	C₂	semidirect
5. #13131059	C₂	semidirect
21. #11718262	C₂²	semidirect
22. #11718261	C₂²	semidirect
229. C₁₃	#33427	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13131210	2
#13131206	2
#13131177	2
C₂×#11718284	2
#13131059	2
#13123254	2
#33427	13